Continuity of ring *-homomorphisms between C*-algebras

Mark Tomforde

Continuity of ring -homomorphisms between C-algebras

Operator Algebras 2009-05-05 v4 Rings and Algebras

Authors: Mark Tomforde

Abstract

The purpose of this short note is to prove that if $A$ and $B$ are unital C*-algebras and $\phi : A \to B$ is a unital *-preserving ring homomorphism, then $\phi$ is contractive; i.e., $\| \phi (a) \| \leq \| a \|$ for all $a \in A$ . (Note that we do not assume $\phi$ is linear.) We use this result to deduce a number of corollaries as well as characterize the form of such unital *-preserving ring homomorphisms. (This note may be of interest to C*-algebraists as well as algebraists who study noncommutative rings and algebras. It is meant to be accessible to a general mathematician and does not require any prior knowledge of C*-algebras.)

Keywords

c*-algebras c*-algebra commutative ring theory

Cite

@article{arxiv.0810.0422,
  title  = {Continuity of ring *-homomorphisms between C*-algebras},
  author = {Mark Tomforde},
  journal= {arXiv preprint arXiv:0810.0422},
  year   = {2009}
}

Comments

7 pages, Version IV changes: Some small typos corrected. This is the final version, to appear. Version III changes: Proposition 3.9 is strengthened, and an alternate proof of Theorem 3.6 is described in the Acknowledgements. Version II changes: A few comments added